From Dark Radiation to Dark Energy: Unified Cosmological Evolution in K-essence Models

We study a class of Unified Dark Matter (UDM) models based on generalized K-essence, where a single scalar field with non-canonical kinetic terms accounts for dark radiation, dark matter, and dark energy. Starting from the purely kinetic Lagrangian proposed by Scherrer (2004), we extend the analysis to quadratic and exponential scalar potentials and explore their phenomenology. All models are implemented in a modified version of \texttt{Hi\_CLASS} and confronted with data from \textit{Planck} 2018, DESI DR1, and Big Bang Nucleosynthesis. The scenarios reproduce the full sequence of cosmic epochs: an early radiation-like phase, a matter-dominated era, and late-time accelerated expansion. The new models predict slightly higher values of the Hubble constant compared to $Λ$CDM, thereby partially alleviating the respective tensions from $\sim 4.4 σ$ to $\sim 3.4 σ$. The quadratic potential requires an ultralight mass that makes it effectively indistinguishable from the Scherrer solution. Overall, generalized K-essence provides a minimal and observationally viable realization of UDM, offering a unified description of the dark sector with distinctive signatures in both early- and late-time cosmology.